Clifford Algebras, Random Graphs, and Quantum Random Variables George Stacey Staples Southern Illinois University at Edwardsville René Schott Université Henri Poincaré Nancy 1

Clifford Algebras of signature (p,q) Cl p,q is the associative algebra of dimension 2 p+q generated by the collection satisfying These algebras are have inherent connections with quantum probability and graph theory.

Clifford Algebras of signature (p,q) A typical element of Cl p,q can be written in the following form: I.e., elements are linear combinations of scalar multiples of multi-vectors (or blades) indexed by subsets of the n-set [n]={1, 2, …, n}. The geometric product is closely related to the logical XOR operator.

Pauli spin matrices Quantum coin flips Satisfy σ * 2 = σ 0 = Identity Anticommute Generate the “Algebra of Physical Space” Cl 3,0

Examples of Clifford Algebras Quaternions Complex number field Algebra of Physical Space (APS) SpaceTime Algebra (STA) Fermion Fock space Fermion algebra

Motivation: QP & CA Algebraic, geometric, and combinatorial properties of Clifford algebras make them interesting objects of study in a variety of contexts. ● Stochastic processes on Clifford algebras of arbitrary signature Accardi, Applebaum, Hudson, von Waldenfels, and others have contributed to the study of processes on the fermion field. Processes on the finite-dimensional fermion algebra and fermion “toy Fock” space are special cases in the study of stochastic processes on Clifford algebras of arbitrary signature.

Motivation: Random Graphs, QP, & CA ● Using combinatorial properties of Clifford algebras to study random graphs and Markov chains Hashimoto, Hora, and Obata have contributed significant work applying quantum probabilistic methods to the study of random graphs. The adjacency matrix of any simple graph is an observable. Defining the quantum decomposition of the adjacency matrix makes application of quantum probabilistic methods natural. In particular, comb graphs, star graphs, Cayley graphs, Hamming graphs, and distance-regular graphs have been considered with this approach. These graphs are related to notions of independence in QP.

Motivation ● Using combinatorial properties of Clifford algebras to study stochastic processes on Clifford algebras. The next logical step: make use of the combinatorial properties to compute expected numbers of cycles, cycle lengths, etc. of random walks on Clifford algebras.

The Problem How do we count the cycles in a simple graph on 6 vertices? Terms along the diagonal of the k th power of the adjacency matrix reveal the closed k-walks based at each vertex.

The Approach Use the combinatorial properties of Clifford algebras to “sieve out” those walks that revisit a vertex. Vertices should “cancel” when revisited. The adjacency matrix will be replaced with a matrix having entries in a Clifford algebra. Vertices will be labeled with nilpotent or unipotent generators of an abelian subalgebra of a Clifford algebra. Some additional tools are needed for this.

Tools: Abelian unipotent-generated subalgebras Cl n sym is the associative algebra of dimension 2 n generated by the collection satisfying Cl n sym is realized within Cl n,n by writing In terms of quantum random variables:

Cl n nil is realized within Cl 2n,2n by writing where f i + denotes the i th fermion creation operator in the 2n-particle fermion algebra. Tools: Abelian nilpotent-generated subalgebras Cl n nil is the associative algebra of dimension 2 n generated by the collection satisfying Using Pauli matrices: and

The Problem How do we count the cycles in a simple graph on 6 vertices?

The Solution Replace the adjacency matrix with a Clifford adjacency matrix. The matrix entries are elements of Cl 6 nil. The generators commute and square to zero. Note: These generators were denoted by  on the previous slide! Elements along the diagonal of A k now reveal k-cycles based at each vertex!

7 Vertices Q. How many 5-cycles does the following graph contain? The adjacency matrix reveals 10 closed 5-walks based at the pendant vertex.

7 Vertices Q. How many 5-cycles does the following graph contain? The Clifford adjacency matrix reveals that the graph contains 6 5-cycles (ignoring orientation and dividing out the 5 choices of base point for each cycle).

Obtaining the number of 7-cycles in a random graph on 12 vertices requires computing the 7 th power of a 12-by-12 matrix having entries in a 4096-dimensional algebra! Note that within the Clifford algebra context, this requires only 6(12 3 )=10,368 multiplications! A strong argument for a computer based on Clifford architecture.

Another Example

An algebraic probability space Let A denote the algebra generated by the n by n Clifford adjacency matrices with involution * defined by a * =(a  ) . Here,  denotes the dual defined by e i  =e [n]\i, and  denotes the matrix transpose. Let ‹‹u›› denote the sum of the real scalar coefficients in the canonical expansion of u  Cl n nil. Define the norm of a  A by

Remark Allowing the entries of the matrix a to be in an arbitrary Clifford algebra with complex coefficients, the dual a  is defined by linear extension of (βe i )  (βe i )=|β| 2 e [n] applied to each entry of a. The norm of a is given by the same expression as before.

Define One finds φ(a*a)  0 for all a in A, and φ(1 A ) = 1; i.e., φ is a state and ( A, φ) is an algebraic probability space. Each adjacency matrix a  A is a quantum random variable whose m th moment corresponds to the number of m-cycles in a graph on n vertices. Letting X m denote the number of m-cycles in the graph associated with a, An algebraic probability space

Remarks: 1. The Clifford adjacency matrix can be formed for any finite graph (directed, undirected, having multiple edges, loops, etc.). Graphs containing multiple edges require labeling edges as well as vertices with Clifford elements. 2.The method can also be applied to edge-existence matrices associated with random graphs. 3. The method can be applied to stochastic matrices & Markov chains.

A notion of quantum decomposition Let U n, L n, and D n denote, respectively, the algebras of n by n upper triangular matrices, lower triangular matrices, and diagonal matrices with entries in Cl n nil and involution * defined by a * =(a  ) . The state φ and the norm ||a|| are defined on each algebra as before. Hence, the three algebraic probability spaces ( U n,φ), ( L n, φ), and ( D n,φ) are obtained. Now the Clifford adjacency matrix A for any directed graph on n vertices is the sum of three algebraic random variables…

Algebraic probability spaces A = a + + a - + a o  ( U n, φ)  ( L n, φ)  ( D n, φ). Any Clifford matrix A can be decomposed in this manner. When A is the Clifford adjacency matrix of any finite graph, φ(A * A)  0, and φ(A m ) = φ((a + + a - + a o ) m ) coincides with the number of m-cycles in the graph.

Note: Given the edge-probability Clifford adjacency matrix of a random graph on n vertices, φ(a m ) corresponds to the expected number of m-cycles in a random graph.

More Combinatorial Applications The number of Hamiltonian cycles in a graph on n vertices is recovered from the Berezin integral of the trace of A n. In the Clifford algebra context this has complexity O(n 4 ). Hence, the NP-complete Hamiltonian-cycle problem is of polynomial time complexity in the Clifford algebra context. Expected hitting times of specific states in Markov chains can be computed. The number of Euler circuits in a finite graph can be obtained by labeling edges in place of vertices. Applying these methods to partitions, Stirling numbers of the second kind, Bell numbers, and Bessel numbers are recovered. A combinatorial approach to iterated stochastic integrals (in classical probability) is obtained in which the iterated stochastic integral of a process appears as the limit in mean of a sequence of Berezin integrals in an ascending chain of Clifford algebras.

Edge-disjoint cycle decompositions

is canonical projection. is an evaluation. It can also be considered canonical projection. Notation:

Edge-disjoint cycle decompositions

Let n be fixed and consider the abelian nilpotent-generated algebra Cl n nil. Define Partitions and Counting Numbers Then for k between 1 and n inclusive, and Stirling numbers of the 2 nd kind n th Bell number

Partitions and Counting Numbers

Non-overlapping Partitions and Bessel Numbers

Bessel Numbers

Canonical raising operator on Cl 3,3 Canonical lowering operator on Cl 3,3 Operator Calculus and Appell Systems

Clifford Appell Systems In the Clifford algebra context, the operator considered is the canonical lowering operator, . This operator has the property  2 =0, so a sequence of related lowering operators is considered to obtain an Appell system with more than two nonzero elements. This sequence can be depicted graphically.

Clifford Appell Systems

Pictured here is the graph associated with the multiplicative random walk on the basis blades of the Clifford algebra Cl 0,2. Idea: Use properties of random walks on hypercubes to establish limit theorems for random walks on Clifford algebras of arbitrary signature Clifford algebras as directed hypercubes

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